Доказать тождество:

Question

Алгебра

Кирила

12 февраля, 13:31

Доказать тождество:

+3

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Препедигна · Answer 1

(sin t + cos t) ^2 - 1 / (ctg t - sin t*cos t) =

= (sin t) ^2 + 2sin t*cos t + (cos t) ^2 - ((sin t) ^2 + (cos t) ^2) / (ctg t - sin t*cos t) =

= (sin t) ^2 + 2sin t*cos t + (cos t) ^2 - (sin t) ^2 - (cos t) ^2 / (ctg t - sin t*cos t) =

=2sin t*cos t / (cos t/sin t) - sin t*cos t

= 2sin t*cos t / (cos t - (sin t) ^2*cos t) / sin t =

= 2sin t*cos t / (cos t (1 - (sin t) ^2) / sin t =

= 2sin t / (1 - (sin t) ^2) / sin t =

= 2 (sin t) ^2 / (1 - (sin t) ^2) =

=2 (sin t) ^2 / (1 - (sin t) ^2) =

= 2 (sin t) ^2 / ((sin t) ^2 + (cos t) ^2 - (sin t) ^2) =

=2 (sin t) ^2 / (cos t) ^2 = 2 (tg t) ^2